Privileged Coordinates and Nilpotent Approximation for Carnot Manifolds, II. Carnot Coordinates

Abstract

This paper is a sequel of Choi and Ponge (J Dyn Control Syst 25:109–157, 2019) and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold, it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single out a special class of privileged coordinates in which the nilpotent approximation at a given point of a Carnot manifold is given by its tangent group. We call these coordinates Carnot coordinates. Examples of Carnot coordinates include Darboux coordinates on contact manifolds and the canonical coordinates of the first kind of Goodman and Rothschild-Stein. By converting the privileged coordinate of Bellaiche into Carnot coordinates, we obtain an effective construction of Carnot coordinates, which we call e-Carnot coordinates. They form the building block of all systems of Carnot coordinates. On a graded nilpotent Lie group, they are given by the group law of the group. For general Carnot manifolds, they depend smoothly on the base point. Moreover, in Carnot coordinates at a given point, they are osculated in a very precise manner by the group law of the tangent group at the point.